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week3.html
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<div id='write' class = 'is-node'><h1><a name='header-n0' class='md-header-anchor '></a>第3周</h1><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n0"><a class="md-toc-inner" href="#header-n0">第3周</a></span><span class="md-toc-item md-toc-h2" data-ref="n5"><a class="md-toc-inner" href="#header-n5">六、逻辑回归(Logistic Regression)</a></span><span class="md-toc-item md-toc-h3" data-ref="n6"><a class="md-toc-inner" href="#header-n6">6.1 分类问题</a></span><span class="md-toc-item md-toc-h3" data-ref="n31"><a class="md-toc-inner" href="#header-n31">6.2 假说表示</a></span><span class="md-toc-item md-toc-h3" data-ref="n72"><a class="md-toc-inner" href="#header-n72">6.3 判定边界</a></span><span class="md-toc-item md-toc-h3" data-ref="n114"><a class="md-toc-inner" href="#header-n114">6.4 代价函数</a></span><span class="md-toc-item md-toc-h3" data-ref="n193"><a class="md-toc-inner" href="#header-n193">6.5 简化的成本函数和梯度下降</a></span><span class="md-toc-item md-toc-h3" data-ref="n259"><a class="md-toc-inner" href="#header-n259">6.6 高级优化</a></span><span class="md-toc-item md-toc-h3" data-ref="n315"><a class="md-toc-inner" href="#header-n315">6.7 多类别分类：一对多</a></span><span class="md-toc-item md-toc-h2" data-ref="n362"><a class="md-toc-inner" href="#header-n362">七、正则化(Regularization)</a></span><span class="md-toc-item md-toc-h3" data-ref="n363"><a class="md-toc-inner" href="#header-n363">7.1 过拟合的问题</a></span><span class="md-toc-item md-toc-h3" data-ref="n393"><a class="md-toc-inner" href="#header-n393">7.2 代价函数</a></span><span class="md-toc-item md-toc-h3" data-ref="n416"><a class="md-toc-inner" href="#header-n416">7.3 正则化线性回归</a></span><span class="md-toc-item md-toc-h3" data-ref="n440"><a class="md-toc-inner" href="#header-n440">7.4 正则化的逻辑回归模型</a></span></p></div><h2><a name='header-n5' class='md-header-anchor '></a>六、逻辑回归(Logistic Regression)</h2><h3><a name='header-n6' class='md-header-anchor '></a>6.1 分类问题</h3><p>参考文档: 6 - 1 - Classification (8 min).mkv</p><p>在这个以及接下来的几个视频中，开始介绍分类问题。</p><p>在分类问题中，你要预测的变量 y是离散的值，我们将学习一种叫做逻辑回归 (Logistic Regression) 的算法，这是目前最流行使用最广泛的一种学习算法。</p><p>在分类问题中，我们尝试预测的是结果是否属于某一个类（例如正确或错误）。分类问题的例子有：判断一封电子邮件是否是垃圾邮件；判断一次金融交易是否是欺诈；之前我们也谈到了肿瘤分类问题的例子，区别一个肿瘤是恶性的还是良性的。</p><p><img src='media/a77886a6eff0f20f9d909975bb69a7ab.png' alt='' /></p><p>我们从二元的分类问题开始讨论。</p><p>我们将因变量(dependent variable)可能属于的两个类分别称为负向类（negative class）和正向类（positive class），则因变量<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-154-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.71ex" height="4.679ex" viewBox="0 -755.9 2888.9 2014.4" role="img" focusable="false" style="vertical-align: -2.923ex;"><defs><path stroke-width="1" id="E156-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E156-MJMAIN-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path stroke-width="1" id="E156-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E156-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="1" id="E156-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E156-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E156-MJMAIN-2208" x="775" y="0"></use><g transform="translate(1442,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E156-MJMAIN-30" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E156-MJMAIN-2C" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E156-MJMAIN-31" x="945" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-154">y\in { 0,1 \\}</script> ，其中 0 表示负向类，1 表示正向类。</p><p><img src='media/f86eacc2a74159c068e82ea267a752f7.png' alt='' /></p><p><img src='media/e7f9a746894c4c7dfd10cfcd9c84b5f9.png' alt='' /></p><p>如果我们要用线性回归算法来解决一个分类问题，对于分类，y 取值为 0 或者1，但如果你使用的是线性回归，那么假设函数的输出值可能远大于 1，或者远小于0，即使所有训练样本的标签 y都等于 0 或 1。尽管我们知道标签应该取值0 或者1，但是如果算法得到的值远大于1或者远小于0的话，就会感觉很奇怪。所以我们在接下来的要研究的算法就叫做逻辑回归算法，这个算法的性质是：它的输出值永远在0到 1 之间。</p><p>顺便说一下，逻辑回归算法是分类算法，我们将它作为分类算法使用。有时候可能因为这个算法的名字中出现了“回归”使你感到困惑，但逻辑回归算法实际上是一种分类算法，它适用于标签 y 取值离散的情况，如：1 0 0 1。</p><p>在接下来的视频中，我们将开始学习逻辑回归算法的细节。</p><h3><a name='header-n31' class='md-header-anchor '></a>6.2 假说表示</h3><p>参考视频: 6 - 2 - Hypothesis Representation (7 min).mkv</p><p>在这段视频中，我要给你展示假设函数的表达式，也就是说，在分类问题中，要用什么样的函数来表示我们的假设。此前我们说过，希望我们的分类器的输出值在0和1之间，因此，我们希望想出一个满足某个性质的假设函数，这个性质是它的预测值要在0和1之间。</p><p>回顾在一开始提到的乳腺癌分类问题，我们可以用线性回归的方法求出适合数据的一条直线：</p><p><img src='media/29c12ee079c079c6408ee032870b2683.jpg' alt='' /></p><p>根据线性回归模型我们只能预测连续的值，然而对于分类问题，我们需要输出0或1，我们可以预测：</p><p>当<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-150-Frame" tabindex="-1" style="font-size: 100%; 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display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.868ex" height="2.577ex" viewBox="0 -806.1 2526.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E151-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E151-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E151-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E151-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E151-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E151-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E151-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E151-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E151-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E151-MJMAIN-29" x="962" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-150">{h_\theta}\left( x \right)</script>小于0.5时，预测 y=0 。</p><p>对于上图所示的数据，这样的一个线性模型似乎能很好地完成分类任务。假使我们又观测到一个非常大尺寸的恶性肿瘤，将其作为实例加入到我们的训练集中来，这将使得我们获得一条新的直线。</p><p><img src='media/d027a0612664ea460247c8637b25e306.jpg' alt='' /></p><p>这时，再使用0.5作为阀值来预测肿瘤是良性还是恶性便不合适了。可以看出，线性回归模型，因为其预测的值可以超越[0,1]的范围，并不适合解决这样的问题。</p><p>我们引入一个新的模型，逻辑回归，该模型的输出变量范围始终在0和1之间。
逻辑回归模型的假设是： <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-4-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.059ex" height="3.044ex" viewBox="0 -906.7 7345 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E4-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E4-MJMATHI-3B8" d="M35 200Q35 302 74 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y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-3D" x="2804" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMATHI-67" x="3860" y="0"></use><g transform="translate(4507,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMATHI-58" x="1526" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJSZ1-29" x="2378" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-4">h_\theta \left( x \right)=g\left(\theta^{T}X \right)</script>
其中：
X 代表特征向量
g 代表逻辑函数（logistic function）是一个常用的逻辑函数为S形函数（Sigmoid function），公式为： <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-6-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.814ex" height="3.395ex" viewBox="0 -956.9 5517.2 1461.5" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="1" id="E6-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path stroke-width="1" id="E6-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E6-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="1" id="E6-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E6-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E6-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E6-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="1" id="E6-MJMATHI-65" d="M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z"></path><path stroke-width="1" id="E6-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMATHI-67" x="0" y="0"></use><g transform="translate(647,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMATHI-7A" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-29" x="858" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-3D" x="2172" y="0"></use><g transform="translate(2950,0)"><g transform="translate(397,0)"><rect stroke="none" width="2048" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-31" x="1198" y="572"></use><g transform="translate(59,-376)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-2B" x="500" y="0"></use><g transform="translate(904,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMATHI-65" x="0" y="0"></use><g transform="translate(329,204)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMATHI-7A" x="778" y="0"></use></g></g></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-6">g\left( z \right)=\frac{1}{1+{{e}^{-z}}}</script>。</p><p>python代码实现：</p><pre class="md-fences md-end-block" lang="python"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">numpy</span> <span class="cm-keyword">as</span> <span class="cm-variable">np</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">sigmoid</span>(<span class="cm-variable">z</span>):</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword cm-error">return</span> <span class="cm-number">1</span> <span class="cm-operator">/</span> (<span class="cm-number">1</span> <span class="cm-operator">+</span> <span class="cm-variable">np</span>.<span class="cm-property">exp</span>(<span class="cm-operator">-</span><span class="cm-variable">z</span>))</span></pre></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 115px;"></div><div class="CodeMirror-gutters" style="display: none; height: 145px;"></div></div></div></pre><p>该函数的图像为：</p><p><img src='media/1073efb17b0d053b4f9218d4393246cc.jpg' alt='' /></p><p>合起来，我们得到逻辑回归模型的假设：</p><p>对模型的理解： <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-6-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.814ex" height="3.395ex" viewBox="0 -956.9 5517.2 1461.5" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="1" id="E6-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path stroke-width="1" id="E6-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E6-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="1" id="E6-MJMAIN-29" 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y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-29" x="858" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-3D" x="2172" y="0"></use><g transform="translate(2950,0)"><g transform="translate(397,0)"><rect stroke="none" width="2048" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-31" x="1198" y="572"></use><g transform="translate(59,-376)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-2B" x="500" y="0"></use><g transform="translate(904,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMATHI-65" x="0" y="0"></use><g transform="translate(329,204)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMATHI-7A" x="778" y="0"></use></g></g></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-6">g\left( z \right)=\frac{1}{1+{{e}^{-z}}}</script>。</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-11-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.868ex" height="2.577ex" viewBox="0 -806.1 2526.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E11-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E11-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E11-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" 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0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path stroke-width="1" id="E8-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 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stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMAIN-3D" x="2804" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMATHI-50" x="3860" y="0"></use><g transform="translate(4778,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMATHI-79" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMAIN-3D" x="1164" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMAIN-31" x="2221" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMAIN-7C" x="2721" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMATHI-78" x="3000" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMAIN-3B" x="3572" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMATHI-3B8" x="4017" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMAIN-29" x="4487" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-8">h_\theta \left( x \right)=P\left( y=1|x;\theta \right)</script>
例如，如果对于给定的x，通过已经确定的参数计算得出<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-11-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.868ex" height="2.577ex" viewBox="0 -806.1 2526.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E11-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E11-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E11-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E11-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E11-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMAIN-29" x="962" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-11">h_\theta \left( x \right)</script>=0.7，则表示有70%的几率y为正向类，相应地y为负向类的几率为1-0.7=0.3。</p><h3><a name='header-n72' class='md-header-anchor '></a>6.3 判定边界</h3><p>参考视频: 6 - 3 - Decision Boundary (15 min).mkv</p><p>现在讲下决策边界(decision boundary)的概念。这个概念能更好地帮助我们理解逻辑回归的假设函数在计算什么。</p><p><img src='media/6590923ac94130a979a8ca1d911b68a3.png' alt='' /></p><p>在逻辑回归中，我们预测：</p><p>当<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-11-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.868ex" height="2.577ex" viewBox="0 -806.1 2526.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E11-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E11-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E11-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E11-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E11-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMAIN-29" x="962" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-11">h_\theta \left( x \right)</script>大于等于0.5时，预测 y=1。</p><p>当<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-11-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.868ex" height="2.577ex" viewBox="0 -806.1 2526.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E11-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E11-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E11-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E11-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E11-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMAIN-29" x="962" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-11">h_\theta \left( x \right)</script>小于0.5时，预测 y=0 。</p><p>根据上面绘制出的 S 形函数图像，我们知道当</p><p>z=0 时 g(z)=0.5</p><p>z>0 时 g(z)>0.5</p><p>z<0 时 g(z)<0.5</p><p>又 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-12-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.996ex" height="2.344ex" viewBox="0 -906.7 3442.7 1009.2" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E12-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="1" id="E12-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E12-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E12-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="E12-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E12-MJMATHI-7A" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E12-MJMAIN-3D" x="746" y="0"></use><g transform="translate(1802,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E12-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E12-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E12-MJMATHI-78" x="2870" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-12">z={\theta^{T}}x</script> ，即：
<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-14-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.809ex" height="2.344ex" viewBox="0 -906.7 1640.2 1009.2" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E14-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E14-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="E14-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-54" x="663" y="513"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-78" x="1067" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-14">{\theta^{T}}x</script> 大于等于 0 时，预测 y=1
<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-14-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.809ex" height="2.344ex" viewBox="0 -906.7 1640.2 1009.2" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E14-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E14-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="E14-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-54" x="663" y="513"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-78" x="1067" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-14">{\theta^{T}}x</script> 小于 0 时，预测 y=0</p><p>现在假设我们有一个模型：</p><p><img src='media/58d098bbb415f2c3797a63bd870c3b8f.png' alt='' /></p><p>并且参数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-112-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E113-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E113-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-112">\theta</script> 是向量[-3 1 1]。 则当<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-16-Frame" tabindex="-1" style="font-size: 100%; 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我们可以绘制直线<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-18-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.869ex" height="2.227ex" viewBox="0 -755.9 5110.3 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E18-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 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463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E18-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E18-MJMAIN-31" x="809" y="-213"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E18-MJMAIN-2B" x="1248" y="0"></use><g transform="translate(2249,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E18-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E18-MJMAIN-32" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E18-MJMAIN-3D" x="3553" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E18-MJMAIN-33" x="4609" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-18">{x_1}+{x_2} = 3</script>，这条线便是我们模型的分界线，将预测为1的区域和预测为 0的区域分隔开。</p><p><img src='media/f71fb6102e1ceb616314499a027336dc.jpg' alt='' /></p><p>假使我们的数据呈现这样的分布情况，怎样的模型才能适合呢？</p><p><img src='media/197d605aa74bee1556720ea248bab182.jpg' alt='' /></p><p>因为需要用曲线才能分隔 y=0 的区域和 y=1 的区域，我们需要二次方特征：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-19-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="44.221ex" height="3.044ex" viewBox="0 -906.7 19039.3 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E19-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 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1]，则我们得到的判定边界恰好是圆点在原点且半径为1的圆形。</p><p>我们可以用非常复杂的模型来适应非常复杂形状的判定边界。</p><h3><a name='header-n114' class='md-header-anchor '></a>6.4 代价函数</h3><p>参考视频: 6 - 4 - Cost Function (11 min).mkv</p><p>在这段视频中，我们要介绍如何拟合逻辑回归模型的参数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-112-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E113-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 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y="319"></use></g></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E21-MJMATHI-58" x="2017" y="0"></use></g></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-21">{h_\theta}\left( x \right)=\frac{1}{1+{e^{-\theta^{T}}X}}</script>带入到这样定义了的代价函数中时，我们得到的代价函数将是一个非凸函数（non-convexfunction）。</p><p><img src='media/8b94e47b7630ac2b0bcb10d204513810.jpg' alt='' /></p><p>这意味着我们的代价函数有许多局部最小值，这将影响梯度下降算法寻找全局最小值。</p><p>线性回归的代价函数为：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-22-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="32.768ex" height="5.846ex" viewBox="0 -1459.5 14108.3 2517" role="img" focusable="false" style="vertical-align: -2.456ex;"><defs><path stroke-width="1" id="E22-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E22-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E22-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 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y="572"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMAIN-32" x="84" y="-532"></use></g><g transform="translate(713,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMATHI-3B8" x="815" y="-219"></use></g><g transform="translate(1633,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJSZ1-29" x="1926" y="-1"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMAIN-2212" x="4240" y="0"></use><g transform="translate(5241,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMATHI-79" x="0" y="0"></use><g transform="translate(499,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJSZ1-29" x="6636" y="-1"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMAIN-32" x="10033" y="878"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-22">J\left( \theta \right)=\frac{1}{m}\sum\limits_{i=1}^{m}{\frac{1}{2}{{\left( {h_\theta}\left({x}^{\left( i \right)} \right)-{y}^{\left( i \right)} \right)}^{2}}}</script> 。
我们重新定义逻辑回归的代价函数为：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-23-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="33.847ex" height="5.846ex" viewBox="0 -1459.5 14573 2517" role="img" focusable="false" style="vertical-align: -2.456ex;"><defs><path stroke-width="1" id="E23-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E23-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 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y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMATHI-6F" x="760" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMATHI-73" x="1246" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMATHI-74" x="1715" y="0"></use><g transform="translate(2243,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-28" x="0" y="0"></use><g transform="translate(389,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMATHI-3B8" x="815" y="-219"></use></g><g transform="translate(1564,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-2C" x="3082" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMATHI-79" x="3527" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-29" x="4025" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-33">Cost\left( {h_\theta}\left( x \right),y \right)</script>简化如下：
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380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z"></path><path stroke-width="1" id="E56-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path stroke-width="1" id="E56-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="1" id="E56-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E56-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E56-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E56-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E56-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E56-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="1" id="E56-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E56-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-2C" x="3082" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-79" x="3527" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-29" x="4025" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-3D" x="6936" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-2212" x="7992" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-79" x="8771" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-D7" x="9490" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-6C" x="10491" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-6F" x="10790" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-67" x="11275" y="0"></use><g transform="translate(11922,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-28" x="0" y="0"></use><g transform="translate(389,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-3B8" x="815" y="-219"></use></g><g transform="translate(1564,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-29" x="2916" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-2212" x="15450" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-28" x="16451" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-31" x="16840" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-2212" x="17563" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-79" x="18564" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-29" x="19061" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-D7" x="19673" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-6C" x="20674" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-6F" x="20972" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-67" x="21458" y="0"></use><g transform="translate(22105,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-31" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-2212" x="1112" y="0"></use><g transform="translate(2112,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-3B8" x="815" y="-219"></use></g><g transform="translate(3288,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E56-MJMAIN-29" x="4639" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-56">Cost\left( {h_\theta}\left( x \right),y \right)=-y\times log\left( {h_\theta}\left( x \right) \right)-(1-y)\times log\left( 1-{h_\theta}\left( x \right) \right)</script>
带入代价函数得到：
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y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-68">J\left( \theta \right)=-\frac{1}{m}\sum\limits_{i=1}^{m}{[{{y}^{(i)}}\log \left( {h_\theta}\left( {{x}^{(i)}} \right) \right)+\left( 1-{{y}^{(i)}} \right)\log \left( 1-{h_\theta}\left( {{x}^{(i)}} \right) \right)]}</script></p><p>Python代码实现：</p><pre class="md-fences md-end-block" lang="python"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">numpy</span> <span class="cm-keyword">as</span> <span class="cm-variable">np</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">cost</span>(<span class="cm-variable">theta</span>, <span class="cm-variable">X</span>, <span class="cm-variable">y</span>):</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable cm-error">theta</span> = <span class="cm-variable">np</span>.<span class="cm-property">matrix</span>(<span class="cm-variable">theta</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">X</span> = <span class="cm-variable">np</span>.<span class="cm-property">matrix</span>(<span class="cm-variable">X</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable cm-error">y</span> = <span class="cm-variable">np</span>.<span class="cm-property">matrix</span>(<span class="cm-variable">y</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">first</span> = <span class="cm-variable">np</span>.<span class="cm-property">multiply</span>(<span class="cm-operator">-</span><span class="cm-variable">y</span>, <span class="cm-variable">np</span>.<span class="cm-property">log</span>(<span class="cm-variable">sigmoid</span>(<span class="cm-variable">X</span> <span class="cm-error">\</span><span class="cm-operator">*</span> <span class="cm-variable">theta</span>.<span class="cm-property">T</span>)))</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable cm-error">second</span> = <span class="cm-variable">np</span>.<span class="cm-property">multiply</span>((<span class="cm-number">1</span> <span class="cm-operator">-</span> <span class="cm-variable">y</span>), <span class="cm-variable">np</span>.<span class="cm-property">log</span>(<span class="cm-number">1</span> <span class="cm-operator">-</span> <span class="cm-variable">sigmoid</span>(<span class="cm-variable">X</span> <span class="cm-error">\</span><span class="cm-operator">*</span> <span class="cm-variable">theta</span>.<span class="cm-property">T</span>)))</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-variable">np</span>.<span class="cm-property">sum</span>(<span class="cm-variable">first</span> <span class="cm-operator">-</span> <span class="cm-variable">second</span>) <span class="cm-operator">/</span> (<span class="cm-builtin">len</span>(<span class="cm-variable">X</span>))</span></pre></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 230px;"></div><div class="CodeMirror-gutters" style="display: none; height: 260px;"></div></div></div></pre><p>在得到这样一个代价函数以后，我们便可以用梯度下降算法来求得能使代价函数最小的参数了。算法为：</p><p>Repeat {
<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-37-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="19.627ex" height="3.978ex" viewBox="0 -1007.2 8450.6 1712.8" role="img" focusable="false" style="vertical-align: -1.639ex;"><defs><path stroke-width="1" id="E37-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E37-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path><path stroke-width="1" id="E37-MJMAIN-3A" d="M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="1" id="E37-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E37-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" 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354 183 323T150 221Q132 149 132 116Q132 21 232 21Q244 21 250 22Q327 35 374 112Q389 137 409 196T430 306Z"></path><path stroke-width="1" id="E37-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E37-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E37-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E37-MJMAIN-2212" x="3557" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E37-MJMATHI-3B1" x="4557" y="0"></use><g transform="translate(5198,0)"><g transform="translate(120,0)"><rect stroke="none" width="1130" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E37-MJMAIN-2202" x="515" y="594"></use><g transform="translate(60,-411)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E37-MJMAIN-2202" x="0" y="0"></use><g transform="translate(401,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E37-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E37-MJMATHI-6A" x="663" y="-213"></use></g></g></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E37-MJMATHI-4A" x="6568" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E37-MJMAIN-28" x="7202" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E37-MJMATHI-3B8" x="7591" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E37-MJMAIN-29" x="8061" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-37">\theta_j := \theta_j - \alpha \frac{\partial}{\partial\theta_j} J(\theta)</script>
(simultaneously update all )
}</p><p>求导后得到：</p><p>Repeat {
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(simultaneously update all )
}</p><p>在这个视频中，我们定义了单训练样本的代价函数，凸性分析的内容是超出这门课的范围的，但是可以证明我们所选的代价值函数会给我们一个凸优化问题。代价函数J(<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-112-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E113-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use 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y="-1"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJSZ1-29" x="5741" y="-1"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-5D" x="20500" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-68">J\left( \theta \right)=-\frac{1}{m}\sum\limits_{i=1}^{m}{[{{y}^{(i)}}\log \left( {h_\theta}\left( {{x}^{(i)}} \right) \right)+\left( 1-{{y}^{(i)}} \right)\log \left( 1-{h_\theta}\left( {{x}^{(i)}} \right) \right)]}</script>
考虑：
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则：
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height="60" x="0" y="220"></rect><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-31" x="2118" y="572"></use><g transform="translate(60,-664)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-2B" x="500" y="0"></use><g transform="translate(904,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMATHI-65" x="0" y="0"></use><g transform="translate(329,204)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-2212" x="0" y="0"></use><g transform="translate(389,0)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMATHI-54" x="469" y="319"></use></g><g transform="translate(1026,0)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMATHI-78" x="0" y="0"></use><g transform="translate(286,179)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-29" x="735" y="0"></use></g></g></g></g></g></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJSZ2-29" x="5910" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-43">={{y}^{(i)}}\log \left( \frac{1}{1+{{e}^{-{\theta^T}{{x}^{(i)}}}}} \right)+\left( 1-{{y}^{(i)}} \right)\log \left( 1-\frac{1}{1+{{e}^{-{\theta^T}{{x}^{(i)}}}}} \right)</script>
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transform="translate(754,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E44-MJMATHI-78" x="0" y="0"></use><g transform="translate(404,256)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E44-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E44-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E44-MJMAIN-29" x="735" y="0"></use></g></g></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E44-MJSZ2-29" x="4680" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-44">=-{{y}^{(i)}}\log \left( 1+{{e}^{-{\theta^T}{{x}^{(i)}}}} \right)-\left( 1-{{y}^{(i)}} \right)\log \left( 1+{{e}^{{\theta^T}{{x}^{(i)}}}} \right)</script></p><p>所以：
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transform="translate(4507,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E149-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E149-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E149-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E149-MJMATHI-58" x="1526" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E149-MJSZ1-29" x="2378" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-148">{h_\theta}\left( x \right)=g\left( {\theta^T}X \right)</script>与线性回归中不同，所以实际上是不一样的。另外，在运行梯度下降算法之前，进行特征缩放依旧是非常必要的。</p><p>一些梯度下降算法之外的选择：
除了梯度下降算法以外，还有一些常被用来令代价函数最小的算法，这些算法更加复杂和优越，而且通常不需要人工选择学习率，通常比梯度下降算法要更加快速。这些算法有：共轭梯度（Conjugate Gradient），局部优化法(Broyden fletcher goldfarb shann,BFGS)和有限内存局部优化法(LBFGS) fminunc是 matlab和octave 中都带的一个最小值优化函数，使用时我们需要提供代价函数和每个参数的求导，下面是 octave 中使用 fminunc 函数的代码示例：</p><pre class="md-fences md-end-block" lang="octave"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><span><span></span>x</span></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">function</span> [<span class="cm-variable">jVal</span>, <span class="cm-variable">gradient</span>] = <span class="cm-variable">costFunction</span>(<span class="cm-variable">theta</span>)</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">jVal</span> = [<span class="cm-error">...</span><span class="cm-variable">code</span> <span class="cm-variable">to</span> <span class="cm-variable">compute</span> <span class="cm-variable">J</span>(<span class="cm-variable">theta</span>)<span class="cm-error">...</span>];</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">gradient</span> = [<span class="cm-error">...</span><span class="cm-variable">code</span> <span class="cm-variable">to</span> <span class="cm-variable">compute</span> <span class="cm-variable">derivative</span> <span class="cm-variable">of</span> <span class="cm-variable">J</span>(<span class="cm-variable">theta</span>)<span class="cm-error">...</span>];</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">end</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">options</span> = <span class="cm-variable">optimset</span>(<span class="cm-string">'GradObj'</span>, <span class="cm-string">'on'</span>, <span class="cm-string">'MaxIter'</span>, <span class="cm-string">'100'</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">initialTheta</span> = <span class="cm-builtin">zeros</span>(<span class="cm-number">2</span>,<span class="cm-number">1</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">[<span class="cm-variable">optTheta</span>, <span class="cm-variable">functionVal</span>, <span class="cm-variable">exitFlag</span>] = <span class="cm-variable">fminunc</span>(<span class="cm-operator">@</span><span class="cm-variable">costFunction</span>, <span class="cm-variable">initialTheta</span>, <span class="cm-variable">options</span>);</span></pre></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 276px;"></div><div class="CodeMirror-gutters" style="display: none; height: 306px;"></div></div></div></pre><p>在下一个视频中，我们会把单训练样本的代价函数的这些理念进一步发展，然后给出整个训练集的代价函数的定义，我们还会找到一种比我们目前用的更简单的写法，基于这些推导出的结果，我们将应用梯度下降法得到我们的逻辑回归算法。</p><h3><a name='header-n193' class='md-header-anchor '></a>6.5 简化的成本函数和梯度下降</h3><p>参考视频: 6 - 5 - Simplified Cost Function and Gradient Descent (10 min).mkv</p><p>在这段视频中，我们将会找出一种稍微简单一点的方法来写代价函数，来替换我们现在用的方法。同时我们还要弄清楚如何运用梯度下降法，来拟合出逻辑回归的参数。因此，听了这节课，你就应该知道如何实现一个完整的逻辑回归算法。</p><p>这就是逻辑回归的代价函数：</p><p><img src='media/eb69baa91c2fc6e7dd8ebdf6c79a6a6f.png' alt='' /></p><p>这个式子可以合并成：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-56-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="63.022ex" height="2.577ex" viewBox="0 -806.1 27134.6 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E56-MJMATHI-43" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 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即，逻辑回归的代价函数：
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x="5741" y="-1"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E57-MJMAIN-5D" x="20500" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-57">=-\frac{1}{m}\sum\limits_{i=1}^{m}{[{{y}^{(i)}}\log \left( {h_\theta}\left( {{x}^{(i)}} \right) \right)+\left( 1-{{y}^{(i)}} \right)\log \left( 1-{h_\theta}\left( {{x}^{(i)}} \right) \right)]}</script>
根据这个代价函数，为了拟合出参数，该怎么做呢？我们要试图找尽量让<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-146-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.758ex" height="2.577ex" viewBox="0 -806.1 2048.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E147-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E147-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E147-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E147-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-4A" x="0" y="0"></use><g transform="translate(800,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-29" x="859" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-146">J\left( \theta \right)</script> 取得最小值的参数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-144-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E145-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E145-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-144">\theta </script>。
<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-60-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.021ex" height="3.861ex" viewBox="0 -806.1 3883.8 1662.6" role="img" focusable="false" style="vertical-align: -1.989ex;"><defs><path stroke-width="1" id="E61-MJMAIN-6D" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path stroke-width="1" id="E61-MJMAIN-69" d="M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z"></path><path stroke-width="1" id="E61-MJMAIN-6E" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path stroke-width="1" id="E61-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E61-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E61-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E61-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E61-MJMAIN-6D"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E61-MJMAIN-69" x="833" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E61-MJMAIN-6E" x="1112" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E61-MJMATHI-3B8" x="945" y="-944"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E61-MJMATHI-4A" x="1835" y="0"></use><g transform="translate(2635,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E61-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E61-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E61-MJMAIN-29" x="859" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-60">\underset{\theta}{\min }J\left( \theta \right)</script>
所以我们想要尽量减小这一项，这将我们将得到某个参数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-144-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E145-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E145-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-144">\theta </script>。
如果我们给出一个新的样本，假如某个特征 x，我们可以用拟合训练样本的参数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-144-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E145-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E145-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-144">\theta </script>，来输出对假设的预测。
另外，我们假设的输出，实际上就是这个概率值：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-63-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.585ex" height="2.577ex" viewBox="-38.5 -806.1 5418.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.089ex;"><defs><path stroke-width="1" id="E64-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 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type="math/tex" id="MathJax-Element-144">\theta </script>为参数，y=1 的概率，你可以认为我们的假设就是估计 y=1 的概率，所以，接下来就是弄清楚如何最大限度地最小化代价函数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-146-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.758ex" height="2.577ex" viewBox="0 -806.1 2048.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E147-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E147-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 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415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E145-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-144">\theta </script>的函数，这样我们才能为训练集拟合出参数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-144-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E145-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E145-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-144">\theta </script>。</p><p>最小化代价函数的方法，是使用梯度下降法(gradient descent)。这是我们的代价函数：
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xlink:href="#E69-MJMATHI-3B8" x="815" y="-219"></use></g><g transform="translate(3357,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJSZ1-29" x="1926" y="-1"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJSZ1-29" x="5741" y="-1"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-5D" x="20500" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-68">J\left( \theta \right)=-\frac{1}{m}\sum\limits_{i=1}^{m}{[{{y}^{(i)}}\log \left( {h_\theta}\left( {{x}^{(i)}} \right) \right)+\left( 1-{{y}^{(i)}} \right)\log \left( 1-{h_\theta}\left( {{x}^{(i)}} \right) \right)]}</script></p><p>如果我们要最小化这个关于<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-112-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E113-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E113-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-112">\theta</script>的函数值，这就是我们通常用的梯度下降法的模板。</p><p><img src='media/171031235527.png' alt='Want ${{\min }_\theta}J(\theta )$：' /></p><p></p><p>我们要反复更新每个参数，用这个式子来更新，就是用它自己减去学习率 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-113-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.41ex" viewBox="0 -504.6 640.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E114-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E114-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-113">\alpha</script>
乘以后面的微分项。求导后得到：</p><p><img src='media/171031235719.png' alt='Want ：' /></p><p>如果你计算一下的话，你会得到这个等式：
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我把它写在这里，将后面这个式子，在 i=1 到 m 上求和，其实就是预测误差乘以<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-72-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.409ex" height="3.745ex" viewBox="0 -1107.7 1467.6 1612.3" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="1" id="E73-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E73-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E73-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E73-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E73-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,521)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-29" x="734" y="0"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-6A" x="809" y="-430"></use></g></svg></span><script type="math/tex" id="MathJax-Element-72">x_j^{(i)}</script> ，所以你把这个偏导数项<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-99-Frame" tabindex="-1" style="font-size: 100%; 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transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E75-MJMAIN-29" x="734" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-74">{\theta_j}:={\theta_j}-\alpha \frac{1}{m}\sum\limits_{i=1}^{m}{({h_\theta}({{x}^{(i)}})-{{y}^{(i)}}){x_{j}}^{(i)}}</script></p><p>所以，如果你有 n 个特征，也就是说：<img src='media/0171031235044.png' alt='' />，参数向量<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-144-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E145-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E145-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-144">\theta </script>包括<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-151-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.461ex" viewBox="0 -806.1 923.4 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E152-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E152-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMAIN-30" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-151">{\theta_{0}}</script> <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-109-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.344ex" viewBox="0 -806.1 923.4 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E110-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E110-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E110-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E110-MJMAIN-31" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-109">{\theta_{1}}</script> <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-110-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.344ex" viewBox="0 -806.1 923.4 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E111-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E111-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" 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y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-88">{\theta_{n}}</script>，那么你就需要用这个式子：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-80-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="37.563ex" height="5.846ex" viewBox="0 -1459.5 16172.8 2517" role="img" focusable="false" style="vertical-align: -2.456ex;"><defs><path stroke-width="1" id="E81-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E81-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 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x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E83-MJMAIN-32" x="663" y="-213"></use></g><g transform="translate(14383,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E83-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E83-MJMAIN-32" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E83-MJMAIN-2B" x="15410" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E83-MJMAIN-2E" x="16188" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E83-MJMAIN-2E" x="16633" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E83-MJMAIN-2E" x="17079" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E83-MJMAIN-2B" x="17524" y="0"></use><g transform="translate(18302,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" 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621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="E84-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJMAIN-3D" x="2804" y="0"></use><g transform="translate(3582,0)"><g transform="translate(397,0)"><rect stroke="none" width="2877" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJMAIN-31" x="1784" y="572"></use><g transform="translate(60,-608)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJMAIN-2B" x="500" y="0"></use><g transform="translate(904,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJMATHI-65" x="0" y="0"></use><g transform="translate(329,204)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJMAIN-2212" x="0" y="0"></use><g transform="translate(389,0)"><use 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style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.344ex" viewBox="0 -806.1 923.4 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E111-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E111-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E111-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E111-MJMAIN-32" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-110">{\theta_{2}}</script> 一直到<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-88-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.309ex" height="2.461ex" viewBox="0 -806.1 994.1 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E89-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E89-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E89-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E89-MJMATHI-6E" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-88">{\theta_{n}}</script>，我们需要用这个表达式来更新这些参数。我们还可以使用 for循环来更新这些参数值，用 <code>for i=1 to n</code>，或者 <code>for i=1 to n+1</code>。当然，不用 for循环也是可以的，理想情况下，我们更提倡使用向量化的实现，可以把所有这些 n个参数同时更新。</p><p>最后还有一点，我们之前在谈线性回归时讲到的特征缩放，我们看到了特征缩放是如何提高梯度下降的收敛速度的，这个特征缩放的方法，也适用于逻辑回归。如果你的特征范围差距很大的话，那么应用特征缩放的方法，同样也可以让逻辑回归中，梯度下降收敛更快。</p><p>就是这样，现在你知道如何实现逻辑回归，这是一种非常强大，甚至可能世界上使用最广泛的一种分类算法。</p><h3><a name='header-n259' class='md-header-anchor '></a>6.6 高级优化</h3><p>参考视频: 6 - 6 - Advanced Optimization (14 min).mkv</p><p>在上一个视频中，我们讨论了用梯度下降的方法最小化逻辑回归中代价函数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-146-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.758ex" height="2.577ex" viewBox="0 -806.1 2048.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E147-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E147-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E147-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E147-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-4A" x="0" y="0"></use><g transform="translate(800,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-29" x="859" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-146">J\left( \theta \right)</script>。在本次视频中，我会教你们一些高级优化算法和一些高级的优化概念，利用这些方法，我们就能够使通过梯度下降，进行逻辑回归的速度大大提高，而这也将使算法更加适合解决大型的机器学习问题，比如，我们有数目庞大的特征量。
现在我们换个角度来看什么是梯度下降，我们有个代价函数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-146-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.758ex" height="2.577ex" viewBox="0 -806.1 2048.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E147-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E147-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E147-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E147-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-4A" x="0" y="0"></use><g transform="translate(800,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-29" x="859" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-146">J\left( \theta \right)</script>，而我们想要使其最小化，那么我们需要做的是编写代码，当输入参数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-112-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E113-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E113-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-112">\theta</script>时，它们会计算出两样东西：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-146-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.758ex" height="2.577ex" viewBox="0 -806.1 2048.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E147-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E147-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E147-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E147-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-4A" x="0" y="0"></use><g transform="translate(800,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-29" x="859" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-146">J\left( \theta \right)</script> 以及J等于 0、1直到 n 时的偏导数项。</p><p><img src='media/394a1d763425c4ecf12f8f98a392067f.png' alt='' /></p><p>假设我们已经完成了可以实现这两件事的代码，那么梯度下降所做的就是反复执行这些更新。
另一种考虑梯度下降的思路是：我们需要写出代码来计算<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-146-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.758ex" height="2.577ex" viewBox="0 -806.1 2048.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E147-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E147-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E147-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E147-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-4A" x="0" y="0"></use><g transform="translate(800,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-29" x="859" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-146">J\left( \theta \right)</script> 和这些偏导数，然后把这些插入到梯度下降中，然后它就可以为我们最小化这个函数。
对于梯度下降来说，我认为从技术上讲，你实际并不需要编写代码来计算代价函数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-146-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.758ex" height="2.577ex" viewBox="0 -806.1 2048.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E147-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E147-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E147-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E147-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-4A" x="0" y="0"></use><g transform="translate(800,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-29" x="859" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-146">J\left( \theta \right)</script>。你只需要编写代码来计算导数项，但是，如果你希望代码还要能够监控这些<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-146-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.758ex" height="2.577ex" viewBox="0 -806.1 2048.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E147-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E147-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E147-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E147-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-4A" x="0" y="0"></use><g transform="translate(800,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-29" x="859" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-146">J\left( \theta \right)</script> 的收敛性，那么我们就需要自己编写代码来计算代价函数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-96-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.371ex" height="2.577ex" viewBox="0 -806.1 1882 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E97-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E97-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E97-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E97-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMATHI-4A" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMAIN-28" x="633" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMAIN-29" x="1492" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-96">J(\theta)</script>和偏导数项<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-99-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.941ex" height="3.978ex" viewBox="0 -1007.2 3418.9 1712.8" role="img" focusable="false" style="vertical-align: -1.639ex;"><defs><path stroke-width="1" id="E100-MJMAIN-2202" d="M202 508Q179 508 169 520T158 547Q158 557 164 577T185 624T230 675T301 710L333 715H345Q378 715 384 714Q447 703 489 661T549 568T566 457Q566 362 519 240T402 53Q321 -22 223 -22Q123 -22 73 56Q42 102 42 148V159Q42 276 129 370T322 465Q383 465 414 434T455 367L458 378Q478 461 478 515Q478 603 437 639T344 676Q266 676 223 612Q264 606 264 572Q264 547 246 528T202 508ZM430 306Q430 372 401 400T333 428Q270 428 222 382Q197 354 183 323T150 221Q132 149 132 116Q132 21 232 21Q244 21 250 22Q327 35 374 112Q389 137 409 196T430 306Z"></path><path stroke-width="1" id="E100-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E100-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path><path stroke-width="1" id="E100-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E100-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E100-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="1130" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMAIN-2202" x="515" y="594"></use><g transform="translate(60,-411)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMAIN-2202" x="0" y="0"></use><g transform="translate(401,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMATHI-6A" x="663" y="-213"></use></g></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMATHI-4A" x="1370" y="0"></use><g transform="translate(2170,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMAIN-29" x="859" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-99">\frac{\partial }{\partial {\theta_j}}J\left( \theta \right)</script>。所以，在写完能够计算这两者的代码之后，我们就可以使用梯度下降。
然而梯度下降并不是我们可以使用的唯一算法，还有其他一些算法，更高级、更复杂。如果我们能用这些方法来计算代价函数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-146-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.758ex" height="2.577ex" viewBox="0 -806.1 2048.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E147-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E147-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E147-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E147-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-4A" x="0" y="0"></use><g transform="translate(800,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-29" x="859" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-146">J\left( \theta \right)</script>和偏导数项<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-99-Frame" tabindex="-1" style="font-size: 100%; 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display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.41ex" viewBox="0 -504.6 640.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E114-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E114-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-113">\alpha</script>，因此它甚至可以为每次迭代选择不同的学习速率，那么你就不需要自己选择。这些算法实际上在做更复杂的事情，而不仅仅是选择一个好的学习率，所以它们往往最终收敛得远远快于梯度下降，这些算法实际上在做更复杂的事情，不仅仅是选择一个好的学习速率，所以它们往往最终比梯度下降收敛得快多了，不过关于它们到底做什么的详细讨论，已经超过了本门课程的范围。</p><p>实际上，我过去使用这些算法已经很长一段时间了，也许超过十年了，使用得相当频繁，而直到几年前我才真正搞清楚共轭梯度法 BFGS 和 L-BFGS的细节。</p><p>我们实际上完全有可能成功使用这些算法，并应用于许多不同的学习问题，而不需要真正理解这些算法的内环间在做什么，如果说这些算法有缺点的话，那么我想说主要缺点是它们比梯度下降法复杂多了，特别是你最好不要使用
L-BGFS、BFGS这些算法，除非你是数值计算方面的专家。实际上，我不会建议你们编写自己的代码来计算数据的平方根，或者计算逆矩阵，因为对于这些算法，我还是会建议你直接使用一个软件库，比如说，要求一个平方根，我们所能做的就是调用一些别人已经写好用来计算数字平方根的函数。幸运的是现在我们有Octave 和与它密切相关的 MATLAB 语言可以使用。</p><p>Octave 有一个非常理想的库用于实现这些先进的优化算法，所以，如果你直接调用它自带的库，你就能得到不错的结果。我必须指出这些算法实现得好或不好是有区别的，因此，如果你正在你的机器学习程序中使用一种不同的语言，比如如果你正在使用C、C++、Java等等，你可能会想尝试一些不同的库，以确保你找到一个能很好实现这些算法的库。因为在L-BFGS或者等高线梯度的实现上，表现得好与不太好是有差别的，因此现在让我们来说明：如何使用这些算法：</p><p><img src='media/743a769317d584a66509fc394b4e6095.png' alt='' /></p><p>比方说，你有一个含两个参数的问题，这两个参数是<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-151-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.461ex" viewBox="0 -806.1 923.4 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E152-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E152-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMAIN-30" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-151">{\theta_{0}}</script>和<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-109-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.344ex" viewBox="0 -806.1 923.4 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E110-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E110-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E110-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E110-MJMAIN-31" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-109">{\theta_{1}}</script>，因此，通过这个代价函数，你可以得到<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-109-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.344ex" viewBox="0 -806.1 923.4 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E110-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E110-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E110-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E110-MJMAIN-31" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-109">{\theta_{1}}</script>和 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-110-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.344ex" viewBox="0 -806.1 923.4 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E111-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E111-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E111-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E111-MJMAIN-32" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-110">{\theta_{2}}</script>的值，如果你将<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-146-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.758ex" height="2.577ex" viewBox="0 -806.1 2048.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E147-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E147-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E147-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E147-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-4A" x="0" y="0"></use><g transform="translate(800,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-29" x="859" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-146">J\left( \theta \right)</script> 最小化的话，那么它的最小值将是<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-109-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.344ex" viewBox="0 -806.1 923.4 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E110-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E110-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E110-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E110-MJMAIN-31" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-109">{\theta_{1}}</script>等于5 ，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-110-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.344ex" viewBox="0 -806.1 923.4 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E111-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E111-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E111-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E111-MJMAIN-32" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-110">{\theta_{2}}</script>等于5。代价函数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-146-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.758ex" height="2.577ex" viewBox="0 -806.1 2048.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E147-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E147-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E147-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E147-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-4A" x="0" y="0"></use><g transform="translate(800,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-29" x="859" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-146">J\left( \theta \right)</script>的导数推出来就是这两个表达式：</p><p><img src='media/bc8430f507924b63781ab9c9f90cd235.png' alt='' /></p><p>。</p><p>如果我们不知道最小值，但你想要代价函数找到这个最小值，是用比如梯度下降这些算法，但最好是用比它更高级的算法，你要做的就是运行一个像这样的Octave 函数：</p><pre class="md-fences md-end-block" lang="octave"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">function</span> [<span class="cm-variable">jVal</span>, <span class="cm-variable">gradient</span>]=<span class="cm-variable">costFunction</span>(<span class="cm-variable">theta</span>)</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">jVal</span>=(<span class="cm-variable">theta</span>(<span class="cm-number">1</span>)<span class="cm-number">-5</span>)<span class="cm-operator">\^</span><span class="cm-number">2</span><span class="cm-operator">+</span>(<span class="cm-variable">theta</span>(<span class="cm-number">2</span>)<span class="cm-number">-5</span>)<span class="cm-operator">\^</span><span class="cm-number">2</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">gradient</span>=<span class="cm-builtin">zeros</span>(<span class="cm-number">2</span>,<span class="cm-number">1</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">gradient</span>(<span class="cm-number">1</span>)=<span class="cm-number">2</span><span class="cm-operator">\*</span>(<span class="cm-variable">theta</span>(<span class="cm-number">1</span>)<span class="cm-number">-5</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">gradient</span>(<span class="cm-number">2</span>)=<span class="cm-number">2</span><span class="cm-operator">\*</span>(<span class="cm-variable">theta</span>(<span class="cm-number">2</span>)<span class="cm-number">-5</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">end</span></span></pre></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 253px;"></div><div class="CodeMirror-gutters" style="display: none; height: 283px;"></div></div></div></pre><p>这样就计算出这个代价函数，函数返回的第二个值是梯度值，梯度值应该是一个2×1的向量，梯度向量的两个元素对应这里的两个偏导数项，运行这个
costFunction 函数后，你就可以调用高级的优化函数，这个函数叫
fminunc，它表示Octave 里无约束最小化函数。调用它的方式如下：</p><pre class="md-fences md-end-block" lang="octave"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">options</span>=<span class="cm-variable">optimset</span>(<span class="cm-string">'GradObj'</span>,<span class="cm-string">'on'</span>,<span class="cm-string">'MaxIter'</span>,<span class="cm-number">100</span>);</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">initialTheta</span>=<span class="cm-builtin">zeros</span>(<span class="cm-number">2</span>,<span class="cm-number">1</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">[<span class="cm-variable">optTheta</span>, <span class="cm-variable">functionVal</span>, <span class="cm-variable">exitFlag</span>]=<span class="cm-variable">fminunc</span>(<span class="cm-operator">@</span><span class="cm-variable">costFunction</span>, <span class="cm-variable">initialTheta</span>, <span class="cm-variable">options</span>);</span></pre></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 115px;"></div><div class="CodeMirror-gutters" style="display: none; height: 145px;"></div></div></div></pre><p>你要设置几个options，这个 options 变量作为一个数据结构可以存储你想要的options，所以 GradObj 和On，这里设置梯度目标参数为打开(on)，这意味着你现在确实要给这个算法提供一个梯度，然后设置最大迭代次数，比方说100，我们给出一个<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-112-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E113-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E113-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-112">\theta</script>的猜测初始值，它是一个2×1的向量，那么这个命令就调用fminunc，这个@符号表示指向我们刚刚定义的costFunction 函数的指针。如果你调用它，它就会使用众多高级优化算法中的一个，当然你也可以把它当成梯度下降，只不过它能自动选择学习速率<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-113-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.41ex" viewBox="0 -504.6 640.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E114-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E114-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-113">\alpha</script>，你不需要自己来做。然后它会尝试使用这些高级的优化算法，就像加强版的梯度下降法，为你找到最佳的<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-114-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E115-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E115-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-114">{\theta}</script>值。</p><p>让我告诉你它在 Octave 里什么样：</p><p><img src='media/bd074e119a52163691cff93c3f42a1ee.png' alt='' /></p><p>所以我写了这个关于theta的 costFunction 函数，它计算出代价函数 jval以及梯度gradient，gradient 有两个元素，是代价函数对于theta(1) 和 theta(2)这两个参数的偏导数。</p><p>我希望你们从这个幻灯片中学到的主要内容是：写一个函数，它能返回代价函数值、梯度值，因此要把这个应用到逻辑回归，或者甚至线性回归中，你也可以把这些优化算法用于线性回归，你需要做的就是输入合适的代码来计算这里的这些东西。</p><p>现在你已经知道如何使用这些高级的优化算法，有了这些算法，你就可以使用一个复杂的优化库，它让算法使用起来更模糊一点。因此也许稍微有点难调试，不过由于这些算法的运行速度通常远远超过梯度下降。</p><p>所以当我有一个很大的机器学习问题时，我会选择这些高级算法，而不是梯度下降。有了这些概念，你就应该能将逻辑回归和线性回归应用于更大的问题中，这就是高级优化的概念。</p><p>在下一个视频，我想要告诉你如何修改你已经知道的逻辑回归算法，然后使它在多类别分类问题中也能正常运行。</p><h3><a name='header-n315' class='md-header-anchor '></a>6.7 多类别分类：一对多</h3><p>参考视频: 6 - 7 - Multiclass Classification_ One-vs-all (6 min).mkv</p><p>在本节视频中，我们将谈到如何使用逻辑回归 (logistic regression)来解决多类别分类问题，具体来说，我想通过一个叫做"一对多" (one-vs-all) 的分类算法。</p><p>先看这样一些例子。</p><p>第一个例子：假如说你现在需要一个学习算法能自动地将邮件归类到不同的文件夹里，或者说可以自动地加上标签，那么，你也许需要一些不同的文件夹，或者不同的标签来完成这件事，来区分开来自工作的邮件、来自朋友的邮件、来自家人的邮件或者是有关兴趣爱好的邮件，那么，我们就有了这样一个分类问题：其类别有四个，分别用y=1、y=2、y=3、y=4来代表。</p><p>第二个例子是有关药物诊断的，如果一个病人因为鼻塞来到你的诊所，他可能并没有生病，用y=1 这个类别来代表；或者患了感冒，用 y=2 来代表；或者得了流感用y=3来代表。</p><p>第三个例子：如果你正在做有关天气的机器学习分类问题，那么你可能想要区分哪些天是晴天、多云、雨天、或者下雪天，对上述所有的例子，y可以取一个很小的数值，一个相对"谨慎"的数值，比如1到3、1到4或者其它数值，以上说的都是多类分类问题，顺便一提的是，对于下标是0 1 2 3，还是 1 2 3 4 都不重要，我更喜欢将分类从 1 开始标而不是0，其实怎样标注都不会影响最后的结果。</p><p>然而对于之前的一个，二元分类问题，我们的数据看起来可能是像这样：</p><p><img src='media/68f56679a2113c7857ab9dd2afebcba8.png' alt='' /></p><p>对于一个多类分类问题，我们的数据集或许看起来像这样：</p><p><img src='media/54d7903564b4416305b26f6ff2e13c04.png' alt='' /></p><p>我用三种不同的符号来代表三个类别，问题就是给出三个类型的数据集，我们如何得到一个学习算法来进行分类呢？</p><p>我们现在已经知道如何进行二元分类，可以使用逻辑回归，对于直线或许你也知道，可以将数据集一分为二为正类和负类。用一对多的分类思想，我们可以将其用在多类分类问题上。</p><p>下面将介绍如何进行一对多的分类工作，有时这个方法也被称为"一对余"方法。</p><p><img src='media/450a83c67732d254dbac2aeeb8ab910c.png' alt='' /></p><p>现在我们有一个训练集，好比上图表示的有三个类别，我们用三角形表示 y=1，方框表示y=2，叉叉表示 y=3。我们下面要做的就是使用一个训练集，将其分成三个二元分类问题。</p><p>我们先从用三角形代表的类别1开始，实际上我们可以创建一个，新的"伪"训练集，类型2和类型3定为负类，类型1设定为正类，我们创建一个新的训练集，如下图所示的那样，我们要拟合出一个合适的分类器。</p><p><img src='media/b72863ce7f85cd491e5b940924ef5a5f.png' alt='' /></p><p>这里的三角形是正样本，而圆形代表负样本。可以这样想，设置三角形的值为1，圆形的值为0，下面我们来训练一个标准的逻辑回归分类器，这样我们就得到一个正边界。</p><p>为了能实现这样的转变，我们将多个类中的一个类标记为正向类（y=1），然后将其他所有类都标记为负向类，这个模型记作<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-115-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.199ex" height="3.511ex" viewBox="0 -1107.7 3099.4 1511.8" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E116-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E116-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E116-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E116-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E116-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E116-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E116-MJMATHI-68" x="0" y="0"></use><g transform="translate(576,521)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E116-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E116-MJMAIN-31" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E116-MJMAIN-29" x="890" y="0"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E116-MJMATHI-3B8" x="815" y="-474"></use><g transform="translate(1747,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E116-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E116-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E116-MJMAIN-29" x="962" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-115">h_\theta^{\left( 1 \right)}\left( x \right)</script>。接着，类似地第我们选择另一个类标记为正向类（y=2），再将其它类都标记为负向类，将这个模型记作 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-116-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.199ex" height="3.511ex" viewBox="0 -1107.7 3099.4 1511.8" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E117-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E117-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E117-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="E117-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E117-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E117-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E117-MJMATHI-68" x="0" y="0"></use><g transform="translate(576,521)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E117-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E117-MJMAIN-32" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E117-MJMAIN-29" x="890" y="0"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E117-MJMATHI-3B8" x="815" y="-474"></use><g transform="translate(1747,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E117-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E117-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E117-MJMAIN-29" x="962" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-116">h_\theta^{\left( 2 \right)}\left( x \right)</script>,依此类推。
最后我们得到一系列的模型简记为： <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-117-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="22.566ex" height="3.511ex" viewBox="0 -1107.7 9715.8 1511.8" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E118-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E118-MJMAIN-28" d="M94 250Q94 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371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path stroke-width="1" id="E119-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="1" id="E119-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 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y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-118">i=\left( 1,2,3....k \right)</script> </p><p>最后，在我们需要做预测时，我们将所有的分类机都运行一遍，然后对每一个输入变量，都选择最高可能性的输出变量。</p><p>总之，我们已经把要做的做完了，现在要做的就是训练这个逻辑回归分类器：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-120-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.944ex" height="3.511ex" viewBox="0 -1107.7 2989.8 1511.8" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E121-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 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min).mkv</p><p>到现在为止，我们已经学习了几种不同的学习算法，包括线性回归和逻辑回归，它们能够有效地解决许多问题，但是当将它们应用到某些特定的机器学习应用时，会遇到过度拟合(over-fitting)的问题，可能会导致它们效果很差。</p><p>在这段视频中，我将为你解释什么是过度拟合问题，并且在此之后接下来的几个视频中，我们将谈论一种称为正则化(regularization)的技术，它可以改善或者减少过度拟合问题。</p><p>如果我们有非常多的特征，我们通过学习得到的假设可能能够非常好地适应训练集（代价函数可能几乎为0），但是可能会不能推广到新的数据。</p><p>下图是一个回归问题的例子：</p><p><img src='media/72f84165fbf1753cd516e65d5e91c0d3.jpg' alt='' /></p><p>第一个模型是一个线性模型，欠拟合，不能很好地适应我们的训练集；第三个模型是一个四次方的模型，过于强调拟合原始数据，而丢失了算法的本质：预测新数据。我们可以看出，若给出一个新的值使之预测，它将表现的很差，是过拟合，虽然能非常好地适应我们的训练集但在新输入变量进行预测时可能会效果不好；而中间的模型似乎最合适。</p><p>分类问题中也存在这样的问题：</p><p><img src='media/be39b497588499d671942cc15026e4a2.jpg' alt='' /></p><p>就以多项式理解，x的次数越高，拟合的越好，但相应的预测的能力就可能变差。</p><p>问题是，如果我们发现了过拟合问题，应该如何处理？</p><ol start='' ><li>丢弃一些不能帮助我们正确预测的特征。可以是手工选择保留哪些特征，或者使用一些模型选择的算法来帮忙（例如PCA）</li><li>正则化。 保留所有的特征，但是减少参数的大小（magnitude）。</li></ol><h3><a name='header-n393' class='md-header-anchor '></a>7.2 代价函数</h3><p>参考视频: 7 - 2 - Cost Function (10 min).mkv</p><p>上面的回归问题中如果我们的模型是：
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\right)={\theta_{0}}+{\theta_{1}}{x_{1}}+{\theta_{2}}{x_{2}}+{\theta_{3}}{x_{3}}+{\theta_{4}}{x_{4}}</script>
我们可以从之前的事例中看出，正是那些高次项导致了过拟合的产生，所以如果我们能让这些高次项的系数接近于0的话，我们就能很好的拟合了。
所以我们要做的就是在一定程度上减小这些参数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-144-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E145-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E145-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-144">\theta </script> 的值，这就是正则化的基本方法。我们决定要减少<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-131-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.461ex" viewBox="0 -806.1 923.4 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E132-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E132-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E132-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E132-MJMAIN-33" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-131">{\theta_{3}}</script>和<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-132-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.344ex" viewBox="0 -806.1 923.4 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E133-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E133-MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMAIN-34" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-132">{\theta_{4}}</script>的大小，我们要做的便是修改代价函数，在其中<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-131-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.461ex" viewBox="0 -806.1 923.4 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E132-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E132-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E132-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E132-MJMAIN-33" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-131">{\theta_{3}}</script>和<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-132-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.344ex" viewBox="0 -806.1 923.4 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E133-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E133-MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMAIN-34" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-132">{\theta_{4}}</script> 设置一点惩罚。这样做的话，我们在尝试最小化代价时也需要将这个惩罚纳入考虑中，并最终导致选择较小一些的<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-131-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.461ex" viewBox="0 -806.1 923.4 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E132-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E132-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E132-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E132-MJMAIN-33" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-131">{\theta_{3}}</script>和<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-132-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.344ex" viewBox="0 -806.1 923.4 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E133-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E133-MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMAIN-34" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-132">{\theta_{4}}</script>。
修改后的代价函数如下：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-130-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="51.932ex" height="5.846ex" viewBox="0 -1459.5 22359.6 2517" role="img" focusable="false" style="vertical-align: -2.456ex;"><defs><path stroke-width="1" id="E131-MJMAIN-6D" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 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id="MathJax-Element-133">J\left( \theta \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{[({{({h_\theta}({{x}^{(i)}})-{{y}^{(i)}})}^{2}}+\lambda \sum\limits_{j=1}^{n}{\theta_{j}^{2}})]}</script></p><p>其中λ又称为正则化参数（Regularization Parameter）。 注：根据惯例，我们不对<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-151-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.461ex" viewBox="0 -806.1 923.4 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E152-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E152-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMAIN-30" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-151">{\theta_{0}}</script> 进行惩罚。经过正则化处理的模型与原模型的可能对比如下图所示：</p><p><img src='media/ea76cc5394cf298f2414f230bcded0bd.jpg' alt='' /></p><p>如果选择的正则化参数λ过大，则会把所有的参数都最小化了，导致模型变成 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-135-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.112ex" height="2.577ex" viewBox="0 -806.1 4784.1 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E136-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E136-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 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151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E136-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E136-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E136-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E136-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E136-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E136-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E136-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E136-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" 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那为什么增加的一项<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-136-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.763ex" height="6.196ex" viewBox="0 -1459.5 4203.7 2667.7" role="img" focusable="false" style="vertical-align: -2.806ex;"><defs><path stroke-width="1" id="E137-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="1" id="E137-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" 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fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E137-MJMATHI-3BB" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E137-MJMAIN-3D" x="861" y="0"></use><g transform="translate(1917,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E137-MJSZ1-2211" x="69" y="0"></use><g transform="translate(0,-890)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E137-MJMATHI-6A" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E137-MJMAIN-3D" x="412" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E137-MJMAIN-31" x="1191" y="0"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E137-MJMATHI-6E" x="545" y="1344"></use></g><g transform="translate(3280,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E137-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E137-MJMAIN-32" x="663" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E137-MJMATHI-6A" x="663" y="-430"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-136">\lambda =\sum\limits_{j=1}^{n}{\theta_j^{2}}</script> 可以使<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-144-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E145-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E145-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-144">\theta </script>的值减小呢？
因为如果我们令λ的值很大的话，为了使Cost Function 尽可能的小，所有的<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-144-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E145-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E145-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-144">\theta </script>的值（不包括<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-151-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.461ex" viewBox="0 -806.1 923.4 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E152-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E152-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMAIN-30" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-151">{\theta_{0}}</script>）都会在一定程度上减小。
但若λ的值太大了，那么<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-144-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E145-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E145-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-144">\theta </script>（不包括<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-151-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.461ex" viewBox="0 -806.1 923.4 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E152-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E152-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMAIN-30" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-151">{\theta_{0}}</script>）都会趋近于0，这样我们所得到的只能是一条平行于x轴的直线。
所以对于正则化，我们要取一个合理的λ的值，这样才能更好的应用正则化。
回顾一下代价函数，为了使用正则化，让我们把这些概念应用到到线性回归和逻辑回归中去，那么我们就可以让他们避免过度拟合了。</p><h3><a name='header-n416' class='md-header-anchor '></a>7.3 正则化线性回归</h3><p>参考视频: 7 - 3 - Regularized Linear Regression (11 min).mkv</p><p>对于线性回归的求解，我们之前推导了两种学习算法：一种基于梯度下降，一种基于正规方程。</p><p>正则化线性回归的代价函数为：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-142-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="43.899ex" height="6.196ex" viewBox="0 -1459.5 18900.9 2667.7" role="img" focusable="false" style="vertical-align: -2.806ex;"><defs><path stroke-width="1" id="E143-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 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type="math/tex" id="MathJax-Element-142">J\left( \theta \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{[({{({h_\theta}({{x}^{(i)}})-{{y}^{(i)}})}^{2}}+\lambda \sum\limits_{j=1}^{n}{\theta _{j}^{2}})]}</script></p><p>如果我们要使用梯度下降法令这个代价函数最小化，因为我们未对进行正则化，所以梯度下降算法将分两种情形：</p><p>.<img src='media/20171101113916.png' alt='' /></p><p>对上面的算法中 j=1,2,...,n 时的更新式子进行调整可得：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-143-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="45.716ex" height="5.846ex" viewBox="0 -1459.5 19683.2 2517" role="img" focusable="false" style="vertical-align: -2.456ex;"><defs><path stroke-width="1" id="E144-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 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可以看出，正则化线性回归的梯度下降算法的变化在于，每次都在原有算法更新规则的基础上令<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-144-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E145-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E145-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-144">\theta </script>值减少了一个额外的值。</p><p>我们同样也可以利用正规方程来求解正则化线性回归模型，方法如下所示：</p><p><img src='media/71d723ddb5863c943fcd4e6951114ee3.png' alt='' /></p><p>图中的矩阵尺寸为 (n+1)*(n+1)。</p><h3><a name='header-n440' class='md-header-anchor '></a>7.4 正则化的逻辑回归模型</h3><p>参考视频: 7 - 4 - Regularized Logistic Regression (9 min).mkv</p><p>针对逻辑回归问题，我们在之前的课程已经学习过两种优化算法：我们首先学习了使用梯度下降法来优化代价函数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-146-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.758ex" height="2.577ex" viewBox="0 -806.1 2048.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E147-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 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286T120 208T113 132Z"></path><path stroke-width="1" id="E147-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-4A" x="0" y="0"></use><g transform="translate(800,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E147-MJMAIN-29" x="859" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-146">J\left( \theta \right)</script>，接下来学习了更高级的优化算法，这些高级优化算法需要你自己设计代价函数<span 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width="1095" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E148-MJMATHI-3BB" x="482" y="584"></use><g transform="translate(60,-376)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E148-MJMAIN-32" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E148-MJMATHI-6D" x="500" y="0"></use></g></g></g><g transform="translate(30128,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E148-MJSZ1-2211" x="69" y="0"></use><g transform="translate(0,-890)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E148-MJMATHI-6A" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E148-MJMAIN-3D" x="412" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E148-MJMAIN-31" x="1191" y="0"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E148-MJMATHI-6E" x="545" y="1344"></use></g><g transform="translate(31491,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E148-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E148-MJMAIN-32" x="663" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E148-MJMATHI-6A" x="663" y="-430"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-147">J\left( \theta \right)=\frac{1}{m}\sum\limits_{i=1}^{m}{[-{{y}^{(i)}}\log \left( {h_\theta}\left( {{x}^{(i)}} \right) \right)-\left( 1-{{y}^{(i)}} \right)\log \left( 1-{h_\theta}\left( {{x}^{(i)}} \right) \right)]}+\frac{\lambda }{2m}\sum\limits_{j=1}^{n}{\theta _{j}^{2}}</script></p><p>Python代码：</p><pre class="md-fences md-end-block" lang="python"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">numpy</span> <span class="cm-keyword">as</span> <span class="cm-variable">np</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">costReg</span>(<span class="cm-variable">theta</span>, <span class="cm-variable">X</span>, <span class="cm-variable">y</span>, <span class="cm-variable">learningRate</span>):</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">theta</span> = <span class="cm-variable">np</span>.<span class="cm-property">matrix</span>(<span class="cm-variable">theta</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">X</span> = <span class="cm-variable">np</span>.<span class="cm-property">matrix</span>(<span class="cm-variable">X</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">y</span> = <span class="cm-variable">np</span>.<span class="cm-property">matrix</span>(<span class="cm-variable">y</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">first</span> = <span class="cm-variable">np</span>.<span class="cm-property">multiply</span>(<span class="cm-operator">-</span><span class="cm-variable">y</span>, <span class="cm-variable">np</span>.<span class="cm-property">log</span>(<span class="cm-variable">sigmoid</span>(<span class="cm-variable">X</span><span class="cm-operator">*</span><span class="cm-variable">theta</span>.<span class="cm-property">T</span>)))</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">second</span> = <span class="cm-variable">np</span>.<span class="cm-property">multiply</span>((<span class="cm-number">1</span> <span class="cm-operator">-</span> <span class="cm-variable">y</span>), <span class="cm-variable">np</span>.<span class="cm-property">log</span>(<span class="cm-number">1</span> <span class="cm-operator">-</span> <span class="cm-variable">sigmoid</span>(<span class="cm-variable">X</span><span class="cm-operator">*</span><span class="cm-variable">theta</span>.<span class="cm-property">T</span>)))</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">reg</span> = (<span class="cm-variable">learningRate</span> <span class="cm-operator">/</span> <span class="cm-number">2</span> <span class="cm-operator">*</span> <span class="cm-builtin">len</span>(<span class="cm-variable">X</span>))<span class="cm-operator">*</span> <span class="cm-variable">np</span>.<span class="cm-property">sum</span>(<span class="cm-variable">np</span>.<span class="cm-property">power</span>(<span class="cm-variable">theta</span>[:,<span class="cm-number">1</span>:<span class="cm-variable">theta</span>.<span class="cm-property">shape</span>[<span class="cm-number">1</span>]],<span class="cm-number">2</span>))</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-variable">np</span>.<span class="cm-property">sum</span>(<span class="cm-variable">first</span> <span class="cm-operator">-</span> <span class="cm-variable">second</span>) <span class="cm-operator">/</span> (<span class="cm-builtin">len</span>(<span class="cm-variable">X</span>)) <span class="cm-operator">+</span> <span class="cm-variable">reg</span></span></pre></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 230px;"></div><div class="CodeMirror-gutters" style="display: none; height: 260px;"></div></div></div></pre><p>要最小化该代价函数，通过求导，得出梯度下降算法为：</p><p><img src='media/99a1373e00809da8acd4de8982f95d91.png' alt='' /></p><p>注：看上去同线性回归一样，但是知道 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-148-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.059ex" height="3.044ex" viewBox="0 -906.7 7345 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E149-MJMATHI-68" d="M137 683Q138 683 209 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Octave 中，我们依旧可以用 fminuc 函数来求解代价函数最小化的参数，值得注意的是参数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-151-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.461ex" viewBox="0 -806.1 923.4 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E152-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E152-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMAIN-30" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-151">{\theta_{0}}</script>的更新规则与其他情况不同。
注意：</p><ol start='' ><li>虽然正则化的逻辑回归中的梯度下降和正则化的线性回归中的表达式看起来一样，但由于两者的<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-150-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.868ex" height="2.577ex" viewBox="0 -806.1 2526.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E151-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E151-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E151-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E151-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E151-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E151-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E151-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E151-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E151-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E151-MJMAIN-29" x="962" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-150">{h_\theta}\left( x \right)</script>不同所以还是有很大差别。</li><li><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-151-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.461ex" viewBox="0 -806.1 923.4 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E152-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E152-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMAIN-30" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-151">{\theta_{0}}</script>不参与其中的任何一个正则化。</li></ol><p>目前大家对机器学习算法可能还只是略懂，但是一旦你精通了线性回归、高级优化算法和正则化技术，坦率地说，你对机器学习的理解可能已经比许多工程师深入了。现在，你已经有了丰富的机器学习知识，目测比那些硅谷工程师还厉害，或者用机器学习算法来做产品。</p><p>接下来的课程中，我们将学习一个非常强大的非线性分类器，无论是线性回归问题，还是逻辑回归问题，都可以构造多项式来解决。你将逐渐发现还有更强大的非线性分类器，可以用来解决多项式回归问题。我们接下来将将学会，比现在解决问题的方法强大N倍的学习算法。</p></div>
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